So I was out drinking last night...how do I remove a tattoo?

Talk to Triss about it. She'll tell you that she can remove it, with the help of a few ingredients:

• Green Mold
• White Myrtle Petals
• Wolf's Aloe

White Myrtle is very common, so you probably already have some. Wolf's Aloe and Green Mold can both be bought from Anezka (the herbalist on the outskirts of town) for 125 orens apiece.

In addition to sjohnston's answer:

You can remove tattoo in Chapter 2 , talk to Sìla. There is the same dialog option for removing the tattoo, and you still need those same ingredients, which can also be bought in Chapter 2.